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Entropy production, viscosity bounds and bumpy black holes

The ratio of shear viscosity to entropy density, $η/s$, is computed in various holographic geometries that break translation invariance (but are isotropic). The shear viscosity does not have a hydrodynamic interpretation in such backgrounds, but does quantify the rate of entropy production due to a strain. Fluctuations of the metric components $δg_{xy}$ are massive about these backgrounds, leading to $η/s < 1/(4π)$ at all finite temperatures (even in Einstein gravity). As the temperature is taken to zero, different behaviors are possible. If translation symmetry breaking is irrelevant in the far IR, then $η/s$ tends to a constant at $T=0$. This constant can be parametrically small. If the translation symmetry is broken in the far IR (which nonetheless develops emergent scale invariance), then $η/s \sim T^{2 ν}$ as $T \to 0$, with $ν\leq 1$ in all cases we have considered. While these results violate simple bounds on $η/s$, we note that they are consistent with a possible bound on the rate of entropy production due to strain.